biorthogonal quantum mechanics...biorthogonal quantum mechanics 5 spoils completeness. a set of...

Biorthogonal Quantum Mechanics

Dorje C. BrodyMathematical Sciences, Brunel University, Uxbridge UB8 3PH, UK

Abstract. The Hermiticity condition in quantum mechanics required for thecharacterisation of (a) physical observables and (b) generators of unitary motionscan be relaxed into a wider class of operators whose eigenvalues are real and whoseeigenstates are complete. In this case, the orthogonality of eigenstates is replaced bythe notion of biorthogonality that defines the relation between the Hilbert space ofstates and its dual space. The resulting quantum theory, which might appropriatelybe called ’biorthogonal quantum mechanics’, is developed here in some detail in thecase for which the Hilbert space dimensionality is finite. Specifically, characterisationsof probability assignment rules, observable properties, pure and mixed states, spinparticles, measurements, combined systems and entanglements, perturbations, anddynamical aspects of the theory are developed. The paper concludes with a briefdiscussion on infinite-dimensional systems.

Submitted to: J. Phys. A: Math. Gen.

1. Introduction

In standard quantum mechanics observable quantities are characterised by Hermitianoperators. The eigenvalues of a Hermitian operator represent possible outcomes of themeasurement of an observable represented by that operator. Once the measurementof, say, the energy is performed and the outcome recorded, the system is in a state ofdefinite energy, that is, there cannot be a transition into another state with a differentenergy. Hermitian operators conveniently encode this feature in the form of theorthogonality of their eigenstates.

The observed lack of transition into another state, however, can only be translatedinto the abstract ‘mathematical’ notion of the orthogonality of states in Hilbertspace via the specification of the probability rules in quantum mechanics. Wheneigenstates of an observable are not orthogonal, however, there is an equally naturalway of assigning probability rules so that the resulting quantum theory appearsidentical to the conventional theory. Evidently, in this case observables are notrepresented by conventional Hermitian operators, since otherwise the eigenstatesare necessarily orthogonal. Nevertheless, if an operator has a complete set ofeigenstates and real eigenvalues, then it becomes a viable candidate for representing aphysical observable. The key mathematical ingredients required to represent physical

Biorthogonal Quantum Mechanics 2

observables are that the eigenvalues are real, and that eigenstates are complete;whereas the notion of orthogonality can be relaxed and substituted by a weakerrequirement of biorthogonality. The resulting quantum theory will thus be calledbiorthogonal quantum mechanics.

There is a substantial literature on the idea of relaxing the Hermiticity requirementfor observables in quantum mechanics. For example, Scholtz et al. [1, 2] proposesthe introduction of a nontrivial metric operator in Hilbert space and defines physicalobservables as self-adjoint operators with respect to the choice of the metric. Viewedfrom the conventional ‘flat’ inner-product structure, therefore, observables are nolonger Hermitian and their eigenstates are not orthogonal, but in the Hilbertspace endowed with this nontrivial metric we recover the ‘standard’ quantumtheory. Bender and others have developed PT-symmetric quantum theory where theHermiticity condition is replaced by the invariance under simultaneous parity andtime reversal operation. A PT-symmetric Hamiltonian is in general not Hermitian,but if the corresponding eigenstates are also PT symmetric, then the eigenvalues arereal and eigenstates may be complete, and can be used to describe quantum systems[3, 4, 5]. Operators that are not Hermitian also play an important role in the physicsof resonance, as discussed, for example, in [6, 7, 8]. The role of biorthogonal systemsin PT-symmetric quantum theories is discussed in Curtright & Mezincescu [9].

The works mentioned here are detailed and substantial, and contain a largenumber of references. In spite of this, here we shall present ‘yet another account’ ofthe subject since a number of basic and foundational ideas of quantum mechanics,already required for the representation of quantum systems modelled on finite-dimensional Hilbert spaces, such as a detailed account of probabilistic interpretations,a characterisation of measurement processes, or a formulation of combined systemsand the role of entanglements, have not been made completely transparent. It turnsout that the approach based from the outset on the use of biorthogonal basis (as in [9])allows us to develop these basic ideas in the most elementary manner. The purposeof the present paper therefore is to develop the formalism of biorthogonal quantummechanics for systems modelled on finite-dimensional Hilbert spaces, and along theway clarify various issues in a transparent and accessible way.

The paper will be organised as follows. We begin in §2 with an overview ofthe biorthogonal system of basis in Hilbert space that arise from the eigenstates ofa complex (i.e. not necessarily Hermitian) Hamiltonian and those of its Hermitianadjoint, for the benefit of readers less acquainted with the material. The effectivenessof the use of biorthogonal basis associated with operators that are not self adjoint hasa long history and goes back to the work of Liouville [10], subsequently developedfurther by Birkhoff [11]. In the case of a real Hilbert space of square-integrablefunctions defined on a finite interval of the real line R, properties of biorthogonalbases associated with operators that are not self adjoint have been worked out indetail by Pell [12, 13]. Many of the results, with suitable modifications, extend intothe complex domain, as developed by Bari [14] (cf. [15]).


In §3 we establish the relation between the Hilbert space H of states and itsdual space H ∗, and this in turn leads to the identification of a consistent probabilityassignment for transitions between states. It will be shown that although eigenstatesof a complex Hamiltonian are not orthogonal inH , they nevertheless do correspondto maximally separated states in the ray-space, hence there cannot be transitionsbetween these states. An analogous conclusion has been drawn previously (e.g., in[4]), but it will become evident that the biorthogonal method employed here leadsto this result in the most elementary fashion, without referring to heavy-handedmathematical arguments. The construction of observables, their expectations, as wellas the notion of general mixed states, are then developed in some detail in §4.

In §5 we discuss measurement-theoretic and further probabilistic aspects ofcomplex Hamiltonians. It will be shown, in particular, that for unitary systemsorthogonality of eigenstates in H is not a condition that can be asserted fromexperiments, thus making any operator having a complete set of eigenstates andreal eigenvalues a viable candidate for the representation of observable quantities.The construction of combined systems in biorthogonal quantum mechanics is thendeveloped in §6, where we also define coherent states in this context. In §7 wedescribe how the Rayleigh-Schrodinger perturbation theory works in the case ofcomplex Hamiltonians. Perturbation of complex Hamiltonians away from eigenstatedegeneracies in fact has been known for some time [16, 17]. The purpose of this sectionis to give a brief review of the idea, partly for completeness and partly on accountof the fact that the result provides an independent confirmation that the probabilityassignment rule of §3 is in some sense the ‘correct’ one. Properties of time evolution ofquantum states generated by a complex Hamiltonian are described in §8, showing thatreality and completeness lead to unitarity, without the orthogonality requirement. In§9 we turn to the discussion of PT-symmetric quantum mechanics, in particular howit ties in with the notion of biorthogonal quantum mechanics. We conclude in §10with a brief discussion towards subtleties arising from the consideration of quantumsystems described by infinite-dimensional Hilbert spaces.

2. Eigenstates of complex Hamiltonians and their adjoints

To begin the analysis of quantum mechanics using basis functions that are in generalnot orthogonal, we shall first review basic properties of eigenstates of generic complexHamiltonians in finite dimensions. Let K = H − iΓ, with H† = H and Γ† = Γ, be acomplex Hamiltonian with eigenstates {|φn〉} and eigenvalues {κn}:

K|φn〉 = κn|φn〉 and 〈φn|K† = κn〈φn|. (1)

We shall assume for now that the eigenvalues {κn} are not degenerate. In addition tothe eigenstates of K, it will be convenient to introduce eigenstates of the Hermitianadjoint matrix K†:

K†|χn〉 = νn|χn〉 and 〈χn|K = νn〈χn|. (2)


Here and in what follows, a ‘Hermitian adjoint’ will be defined by the conventionthat K† denotes the complex-conjugate transpose of K. The reason for introducingthe additional states {|χn〉} is because the eigenstates {|φn〉} of K are in general notorthogonal:

〈φm|φn〉 = 2i〈φm|Γ|φn〉

κm − κn= 2〈φm|H|φn〉

κm + κn(3)

for m , n, which follows from the facts that 2iΓ = K† − K and that 2H = K† + K. Ananalogous result

〈χm|χn〉 = 2i〈χm|Γ|χn〉

νn − νm= 2〈χm|H|χn〉

νn + νm(4)

holds for the eigenstates {|χn〉}of K†. Of course, for a given K some of its eigenstates canbe orthogonal, but if K is not Hermitian, then a typical situation that arises is wherenot all the eigenstates are orthogonal. Hence conventional projection techniquesso commonly used in many calculations of quantum mechanics, for example, inmeasurement theory or perturbation analysis, are ineffective when dealing with theeigenstates of a complex Hamiltonian [16].

With the aid of the conjugate basis {|χn〉}, let us first establish that the eigenstates{|φn〉} of K, although not orthogonal, are nevertheless linearly independent. To showthis, suppose the converse that {|φn〉} are linearly dependent. Then there exists a setof numbers {cn} such that

∑n |cn|

2 , 0, and that∑n

cn|φn〉 = 0. (5)

Transvecting this relation with 〈χm| from the left, we find, for each m, that cm〈χm|φm〉 =

0, where we have made use of the facts that

〈χn|φm〉 = δnm〈χn|φn〉 (6)

and that 〈χn|φn〉 , 0. To see that (6) holds, we note that by definitions (1) and (2) wehave

〈χm|K|φn〉 = νm〈χm|φn〉 = κn〈χm|φn〉. (7)

Hence 〈χm|φn〉 = 0 if κn , νm, and κn = νm if 〈χm|φn〉 , 0. Since 〈χm|φn〉 = 0 cannothold for all {|χm〉}, there has to be at least one νm such that κn = νm. On the otherhand, by assumption the eigenvalues are not degenerate, so there cannot be morethan one νm for which κn = νm. Without loss of generality we can label the states suchthat we have κn = νn for all n. It follows that 〈χm|φn〉 = 0 if n , m but 〈χn|φn〉 , 0,and this establishes (6). Now since 〈χm|φm〉 , 0 when K is nondegenerate, we musthave cm = 0 for all m, contradicting the hypothesis. It follows that the nondegenerateeigenstates {|φn〉} of K are linearly independent, and thus span the Hilbert space H ,since the number of linearly independent basis elements agrees with the Hilbert-spacedimensionality. In other words, {|φn〉} forms a complete set of basis forH . Additionally,they are minimal in that exclusion of any one of the elements |φk〉 from the set {|φn〉}


spoils completeness. A set of basis elements that is both minimal and complete iscalled exact. In finite dimensions, the exactness of {|φn〉} implies the exactness of {|χn〉},whereas in infinite dimensions this no longer is the case, as discussed below in §10.

Using the independence of the states {|φn〉}we can establish the relation:∑n

|φn〉〈χn|

〈χn|φn〉= 1, (8)

which hold in finite dimensions away from degeneracies. To show this, we remarkthat if F has the property that 〈ψ|F|ψ〉 = 〈ψ|ψ〉 holds true for an arbitrary vector |ψ〉,then it must be that F = 1. Writing |ψ〉 =

∑m cm|φm〉 for some {cm}we have

〈ψ|

∑n

|φn〉〈χn|

〈χn|φn〉

|ψ〉 =∑

n

∑m

cmcn〈φm|φn〉 = 〈ψ|ψ〉, (9)

and this establishes the claim.The operator Πn defined by (cf. [18])

Πn =|φn〉〈χn|

〈χn|φn〉(10)

thus plays the role of a projection operator satisfying ΠnΠm = δnmΠn. Although Πn isnot Hermitian, its eigenvalues are all zero, except one which is unity, for which theeigenstate is |φn〉. Writing Φn = |φn〉〈φn|/〈φn|φn〉 for the eigenstate projector we have

ΠnΦn = ΦnΠn = Φn. (11)

It follows, in particular, that

(1 − Πn)|φn〉 = (1 − Π†n)|χn〉 = 0. (12)

While the complex Hamiltonian K does not admit the representation∑

n κnΦn, due tothe fact that ΦnΦm , δnmΦm, it nevertheless can be expressed in the form (cf. [19]):

K =∑

n

κnΠn. (13)

It follows, furthermore, that if we write, for an arbitrary state |ψ〉 =∑

m cm|φm〉,

ψχn =〈φn|ψ〉√〈φn|χn〉

and ψφn =〈χn|ψ〉√〈χn|φn〉

, (14)

then we have

〈ϕ|ψ〉 =∑

n

ϕχnψφn . (15)

A form of this result for real Hilbert-space vectors was obtained in [12].

3. Quantum probabilities

In the foregoing discussion we have not commented on the norm convention. Inquantum theory, the norm of a state is closely related to probabilistic interpretations


of measurement outcomes. Hence we wish to fix our norm convention so that itis consistent with probabilistic considerations of a quantum system when energyeigenstates are not orthogonal. Now in the literature on the use of biorthogonalbasis for complex Hamiltonians, especially in quantum chemistry, the norm of theeigenvectors are often (but not always; cf. [20, 21] for a related discussion) assumedto take values larger than unity so as to ensure the following relation holds for all n:

〈χn|φn〉 = 1. (16)

Under this convention, eigenvectors will no longer be normalised. In particular, if weassume that all eigenstates have the same Hermitian norm so that 〈φn|φn〉 = 〈φm|φm〉

for all n,m, then we have 〈φn|φn〉 ≥ 1. This might at first seem a little odd fromthe viewpoint of traditional Hermitian quantum mechanics, however, for a range ofanalysis that follow, it turns out that the convention 〈χn|φn〉 = 1 leads to considerablesimplifications.

To begin, we recall that in standard quantum mechanics, the ‘transitionprobability’ between a pair of states |ξ〉 and |η〉 is given by the ratio of the form〈ξ|η〉〈η|ξ〉/〈ξ|ξ〉〈η|η〉. Under the convention 〈χn|φn〉 = 1, however, we cannot maintaina consistent probabilistic interpretation from this definition. For instance, if the stateof the system is in an eigenstate |φn〉 of a complex Hamiltonian K, then on accountof stationarity there cannot be a ‘transition’ into another state |φm〉, m , n, eventhough 〈φm|φn〉 , 0; whereas according to the conventional definition the transitionprobability between these states is nonzero. To reconcile these apparent contradictionswe need the introduction of the so-called associated state that defines duality relationsbetween elements of the Hilbert spaceH and its dual spaceH ∗.

For an arbitrary state |ψ〉, we define the associated state |ψ〉 according to thefollowing relations:

|ψ〉 =∑

n

cn|φn〉 ⇔ 〈ψ| =∑

n

cn〈χn| ⇒ |ψ〉 =∑

n

cn|χn〉. (17)

We shall let (17) determine the duality relation on the state space: |ψ〉 ∈ H ⇔ |ψ〉 ∈ H ∗.Putting the matter differently, the state dual to |ψ〉 is given by 〈ψ| of (17); the state|ψ〉 associated to |ψ〉 is then given by the Hermitian conjugate of 〈ψ|. The quantum-mechanical inner product for a biorthogonal system is thus defined as follows: If|ψ〉 =

∑n cn|φn〉 and |ϕ〉 =

∑n dn|φn〉, then

〈ϕ,ψ〉 ≡ 〈ϕ|ψ〉 =∑n,m

dncm〈χn|φm〉 =∑

n

dncn. (18)

Since we demand the convention that 〈χn|φn〉 = 1 for all n, we can assume that

〈ψ|ψ〉 =∑

n

cncn = 1. (19)

It also follows that pn = cncn defines the transition probability between |ψ〉 and |φn〉:

pn =〈χn|ψ〉〈ψ|φn〉

〈ψ|ψ〉〈χn|φn〉, (20)


provided that the Hilbert space pairing is defined by the convention (18). Here fordefiniteness we have expressed pn in a homogeneous form that is invariant undercomplex scale transformations of the states. The interpretation of the number pn is asfollows: if a system is in a state characterised by the vector |ψ〉, and if a measurement isperformed on the ‘complex observable’ K, then the probability that the measurementoutcome taking the value κn is given by pn.

More generally, the overlap distance s between the two states |ξ〉 and |η〉 will bedefined according to the prescription:

cos2 12s =

〈ξ|η〉〈η|ξ〉

〈ξ|ξ〉〈η|η〉. (21)

A short exercise making use of the Cauchy-Schwarz inequality shows that the rightside of (21) is real, nonnegative, and lies between zero and one, thus qualifying therequired probabilistic conditions. In particular, s = 0 only if |ξ〉 = |η〉; whereas s = πonly if

∑n cndn = 0 where |ξ〉 =

∑n cn|φn〉 and |η〉 =

∑n dn|φn〉.

In quantum mechanics the notion of probability is closely related to that ofdistance. To see this, suppose that |η〉 = |ξ〉 + |dξ〉 is a neighbouring state to |ξ〉. Thenexpanding (21) and retaining terms of quadratic order, we obtain the following formof the line element, known as the Fubini-Study line element:

ds2 = 4〈ξ|ξ〉〈dξ|dξ〉 − 〈ξ|dξ〉〈dξ|ξ〉

〈ξ|ξ〉2. (22)

As an illustrative example, consider a two-dimensional Hilbert space spanned by apair of states (|φ1〉, |φ2〉). Then an arbitrary normalised—in the sense of (19)—state |ξ〉can be expressed in the form

|ξ〉 = cos 12θ|φ1〉 + sin 1

2θeiϕ|φ2〉. (23)

Evidently we have 〈ξ|ξ〉 , 1 but 〈ξ|ξ〉 = 1, on account of (16). Taking the differentialof |ξ〉 and substituting the resulting expression in (22), making use of (17), we deducethat the line element is given by

ds2 = 14

(dθ2 + sin2 θdϕ2

). (24)

It follows that the state space defined by the relation 〈ξ|ξ〉 = 1 is a two-sphere ofradius one half—the Bloch sphere of complex Hamiltonian systems. We shall havemore to say about this.

4. Observables and states

We have shown in (13) that a complex Hamiltonian K admits a spectral decompositionin terms of the complex projection operators {Πn}. Evidently, for a fixed biorthogonalbasis {|φn〉, |χn〉} there are uncountably many such (commuting family of) operatorsfor which eigenvalues are entirely real, even though they are not Hermitian in thesense that K† does not agree with K. In fact, the class of such ‘real’ operators in thisspace is wider and contains those that do not commute with the Hamiltonian K.


Given a fixed biorthogonal basis {|φn〉, |χn〉}, a generic operator F can be expressedin the form

F =∑n,m

fnm|φn〉〈χm|. (25)

Note that F can likewise be expressed in terms of the nonorthogonal basis {|φn〉}:

F =∑n,m

ϕnm|φn〉〈φm|, (26)

since the set {|φn〉} is complete. However, in this case the array {ϕnm} cannot be viewedas a matrix, whereas the array { fnm} can, which shows the advantage of the use ofbiorthogonal basis. Thus, if G is another operator with ‘matrix’ elements gnm in thebasis {|φn〉, |χn〉}, then the matrix element of the product FG is just

∑l fnlglm.

If F and G are nondegenerate Hermitian—in the usual sense—operators, theeigenstates of F can always be transformed unitarily into those of G. For complexoperators, however, this is no longer the case. Nevertheless, two operators F andG will be said to belong to the same class of observables if there is a unitarytransformation between the basis of F and G.

The expectation value of a generic observable F in a pure state |ψ〉 is defined bythe expression

〈F〉 =〈ψ|F|ψ〉〈ψ|ψ〉

. (27)

In particular, if the array { fnm} in (25) is ‘biorthogonally Hermitian’ in the sense thatfnm = fmn, then 〈F〉 defined by (27) is real for all states |ψ〉, even though 〈ψ|F|ψ〉/〈ψ|ψ〉is not real for most states. Thus, the notion of Hermiticity extends naturally to thebiorthogonal setup, and we are able to speak about physical observables in the usualsense. This follows from the fact that although F is not Hermitian in the sense thatF† , F, its expectation value (27) in an arbitrary state |ψ〉 is nevertheless real becausethe corresponding matrix { fnm} in the biorthogonal basis is Hermitian. If we let|ψ〉 =

∑n cn|φn〉 and substitute this in (27), making use of (25), then we find

〈F〉 =

∑n,m cncm fnm∑

n cncn. (28)

In particular, if {|φn〉} are eigenstates of F, then we can write fnm = fnδnm, where { fn}

are the eigenvalues of F, hence

〈F〉 =∑

n

pn fn, (29)

which is consistent with our probabilistic interpretation of the biorthogonal system.The matrix interpretation here nevertheless requires further clarification. If a

Hermitian ‘matrix’ fnm is given without the information about the choice of basis, thenthere is no procedure to determine whether F is Hermitian; whereas for orthogonalbases, the data fnm is sufficient to determine whether F is Hermitian, even though the


choice of the orthogonal basis remains arbitrary. To make this transparent, supposethat {|en〉} is an orthonormal basis ofH such that

|φn〉 =∑

k

ukn|ek〉, |χn〉 =

∑k

vkn|ek〉. (30)

Then the matrix element of the observable F in this orthonormal basis is given by

F =∑n,m

∑k,l

fklunk vm

l

|en〉〈em|. (31)

In this way we see more explicitly that while the reality of F merely requiresHermiticity of { fnm}, the Hermiticity of F requires a more stringent condition that∑

k,l

fklunk vm

l =∑

k,l

fklumk vn

l . (32)

In particular, if F is Hermitian so that F† = F, then {|en〉} can be chosen to be |φn〉 so thatun

k = vnk = δn

k and (32) reduces to the familiar condition fnm = fmn; if F is symmetric,then the left side of (32) is invariant under the interchange of indices m↔ n, and wehave vn

k = unk , i.e. components of |χn〉 are complex conjugates of the components of

|φn〉. The expansion coefficients {unk } are unique up to unitary transformations. The

linear independence of {|φn〉} implies that {ukn} is invertible, and the orthonormality

condition 〈χn|φm〉 = δnm implies that the inverse of {ukn} is given by {vk

n}. Phraseddifferently, if we write (30) in the form |φn〉 = u|en〉 and |χn〉 = v|en〉, then we havev†u = 1; if F is real (biorthogonally Hermitian), then

F† = vv† F uu† = (uu†)−1F (uu†), (33)

where uu† is an invertible positive Hermitian operator.As an elementary illustrative example, consider the complex 2 × 2 Hamiltonian

K = σx − iγσz with γ2 < 1. A short calculation shows that the eigenstates of K and K†,in the region γ2 < 1 for which the eigenvalues ±

√1 − γ2 are real, are given by

|φ±〉 = n±

(1

iγ ±√

1 − γ2

), |χ±〉 = n∓

(1

−iγ ±√

1 − γ2

), (34)

where n2±

= (1 ∓ iγ/√

1 − γ2)/2, and where we have written |φ+〉 for |φ1〉, and so on.An arbitrary observable for which the expectation value defined by (27) is real can beexpressed, up to trace, as a linear combination of the deformed Pauli matrices

σγx =1√

1 − γ2

(−iγ 1

1 iγ

), σγy =

(0 −ii 0

), σγz =

1√1 − γ2

(1 iγiγ −1

). (35)

These are obtained according to the prescriptions

σγx = |φ1〉〈χ2| + |φ2〉〈χ1|, σγy = −i|φ1〉〈χ2| + i|φ2〉〈χ1|, σγz = |φ1〉〈χ1| − |φ2〉〈χ2|. (36)

It should be evident that the triplet (σγx , σγy, σ

γz ) fulfils the standard su(2) commutation

relations, and that in the Hermitian limit γ → 0 we recover the standard Pauli


matrices. The expectation values, in the sense of (27), of these Pauli matrices in ageneric state (23) are thus given by

〈σγx〉 = sinθ cosϕ, 〈σγy〉 = sinθ sinϕ, 〈σγz 〉 = cosθ. (37)

Note that the right-sides of these expectation values are independent of γ, on accountof the γ-dependence of the eigenstates. Expectation values of Hermitian operators,such as the usual Pauli matrices, on the other hand, are in general not real since theydo not represent physical observables in the biorthogonal system.

It should be evident, incidentally, that in the case of a two-level system, thechoice of the biorthogonal system {|φ1,2〉} is uniquely determined by the overlapdistance arccos |〈φ1|φ2〉|, up to unitarity. Physical observables constructed under thebiorthogonal system {|φ1,2〉, |χ1,2〉} therefore belong to the same class of observables asthose constructed from another system {|φ′1,2〉, |χ

′

1,2〉}, provided that |〈φ1|φ2〉| = |〈φ′1|φ′

2〉|.We have spoken about pure states thus far, but the state of a physical system in

quantum mechanics is, more generally, and perhaps more commonly, characterisedby a mixed state density matrix:

ρ =∑n,m

ρnm|φn〉〈χm|. (38)

A density matrix ρ is thus not Hermitian in the usual sense so that ρ , ρ†, butit is ‘Hermitian’ with respect to the choice of biorthogonal basis {|φn〉, |χn〉} so thatρnm = ρmn. The eigenvalues of ρ are nonnegative and add up to unity. The expectationvalue of a generic observable (25) in the state ρ is thus defined by

〈F〉 = tr(ρF) =∑

n

〈χn|ρF|φn〉 =∑n,m

ρnm fmn. (39)

It should be evident that a necessary and sufficient condition for the reality of 〈F〉, foran arbitrary ρ, is that fnm = fmn.

A simple example of a density matrix arises if a quantum system described bya complex Hamiltonian K is immersed in a heat bath of inverse temperature β. Inparticular, if the eigenvalues {κn} of K are all real, then after a passage of time thesystem will reach an equilibrium state

ρ =e−βK

tr(e−βK)=

∑n

e−βκn−ln Z(β)|φn〉〈χn|, (40)

if we assume the postulate that an equilibrium state should maximise the vonNeumann entropy − tr(ρ ln ρ) subject to the constraint that the system must possessa definite energy expectation tr(ρK). Here, Z(β) = tr(e−βK) denotes the partitionfunction. The reality of all the eigenvalues of K is crucial for the existence of acanonical distribution (40), owing to properties of the dynamics of the system, asdescribed below in §8.


5. Measurement of spin-12 particle

We now wish to turn to the discussion about the Bloch sphere introduced in §3above, in the context of a spin- 1

2 particle system in quantum mechanics. To this endwe recall first with the general discussion that in standard nonrelativistic quantummechanics, the wave function of a particle splits into two components, one associatedwith its spacial symmetry and the other associated with its internal symmetry (suchas spin, isospin, colour, flavour, etc.). Since in the nonrelativistic context these spacialand internal symmetries are independent, if one is interested only in the internalsymmetry of a particle, then it is a common practice to ignore the spacial degrees offreedom of the wave function (belonging to an infinite-dimensional Hilbert space)and focus attention on the internal symmetries (belonging to a finite-dimensionalHilbert space). It follows, in particular, that internal symmetries of a particle, a priori,do not concern the spacial degrees of freedom.

In spite of the independence of these symmetries, one commonly speaks, forinstance, about the spin of an electron in a certain spacial direction. The reason whythis is permissible has its origin in the mathematical structure of the state space ofa spin- 1

2 particle system: The space of states for this system is a two-sphere—in thequantum context this is often referred to as the Bloch sphere—which can be embeddedin a three-dimensional Euclidean space R3. The implication of this remarkable factis that one may select an arbitrary point on the state space and declare this point tobe, say, the ‘north pole’. In this manner, each spin degrees of freedom of a spin-1

2particle is mapped, one-to-one, to a direction in three dimensions. This identificationis sometimes referred to as the Pauli correspondence, and can be seen in differentways. For example, from (37) one sees that the expectation value of a spin operator(which is one-half of the Pauli matrices) takes a value on a sphere of radius one-half in R3 (see [22, 23, 24] for further discussion on the relation between the spacialdimension of the space-time and the spin of quantum particles).

With this background of standard quantum mechanics in mind, let us now turnto a spin-1

2 particle characterised by a Hamiltonian K whose eigenstates are notorthogonal. The relevant mathematical machineries have already been introducedabove, but let us introduce them here in a slightly different order: Rather than startingfrom a Hamiltonian K, let us start from the specification of the eigenstates. Specifically,suppose that a pair of distinct states (|φ1〉, |φ2〉) is given in a two-dimensional Hilbertspace H such that 〈φ1|φ2〉 , 0. We then find the conjugate pair (|χ1〉, |χ2〉) bysolving the equations 〈χ1|φ2〉 = 0 and 〈χ2|φ1〉 = 0, satisfying the norm convention〈χ1|φ1〉 = 〈χ2|φ2〉 = 1; solutions will be unique up to overall phases. We then identifythe Hamiltonian according to

K = κ1|φ1〉〈χ1| + κ2|φ2〉〈χ2|, (41)

which, alternatively, can be expressed in the form K = B · σ for some choice of realvector B, where σ is the Pauli-matrix vector obtained by use of the biorthogonal basis,in accordance with (36). This Hamiltonian, although not Hermitian, nevertheless has


the interpretation of representing the energy of a spin- 12 particle system immersed in

an external magnetic field B in R3.This result follows from our probability assignment rule (21). To see this, we

recall that a generic state of the particle can be expressed in the form (23). Nowthe spherical coordinates used in (23) show that the two eigenstates |φ1〉 and |φ2〉

are antipodal points on the Bloch sphere, even though they are not orthogonal in H .We have explained that when an experimentalist performs a spin measurement, thedirection of the measurement apparatus in R3 is in one-to-one correspondence withthe point on the Bloch sphere S2, not so much with the direction in Hilbert spaceH assuch, in the chain of abstractionR3

→ S2→H . To put the matter differently, the data

obtained from the Stern-Gerlach experiment (see [25] for a curious historical accountof the experiment) does not provide information concerning whether the ‘spin-up’state and ‘spin-down’ state correspond to orthogonal vectors inH ; it merely tells usthat they correspond to antipodal points on S2, whereas going from S2 toH requiresfurther milages requiring more information than mere experimental data.

For sure the use of orthogonal bases—hence the use of Hermitian operators—simplifies the algebra, but apart from this ‘convenience’ argument, there is no needto require orthogonality inH ; all that is needed is the completeness. We are thereforeled to the following conclusion:

Proposition 1 In finite dimensions, the interrelation, i.e. the overlap distances, of theeigenstates of nondegenerate observables with real eigenvalues in Hilbert space cannot bedetermined from experimental data.

In other words, any operator possessing the relevant eigenvalue structure is alegitimate candidate for a physical observable. Hence Hermitian operators haveno privileged status, apart from their ability in making calculations simpler. Thisconclusion, however, is not necessarily true in infinite dimensions; likewise in finitedimensions, one can identify differences between Hermitian and non-Hermitianobservables if at least one of the eigenvalues is complex, or if there are degeneraciesof eigenstates. We shall have more to say about these points.

6. Spin particles and combined systems

Particles with higher spin numbers can be formulated analogously. Of course,one might ask, even in the case of standard quantum mechanics with Hermitianobservables, in which way spin measurements in R3 can be related to points on thestate space since the dimensionality of the state space for higher spin systems islarger than three and hence it cannot be embedded inR3. The way to realise the Paulicorrespondence for higher spin systems is to note the fact that in the state space foreach spin, there is a family of privileged quantum states, sometimes called the su(2)coherent states, that fully embody information concerning directional data in R3 (see[26, 27] for a detailed discussion), and that the coherent state subspace is always a two


sphere S2 that can be embedded in R3. It is via this device that the idea of the Paulicorrespondence for spin- 1

2 particle can be extended to arbitrary spin particles. To putthe matter differently, for higher spins there is a natural embedding of the directionaldata of R3 in the state space of the system.

It should be evident from the discussion of the preceding section that a similarline of reasoning is applicable to biorthogonal quantum systems. As an example,consider a spin-1

2 state vector |ψ〉 = c1|φ1〉+c2|φ2〉 inH2, normalised as usual accordingto 〈ψ|ψ〉 = 1. We embed this state inH3 by consideration of the product state:

|ψ,ψ〉 = c21|φ1, φ1〉 +

√

2c1c2

(|φ1, φ2〉 + |φ2, φ1〉

√2

)+ c2

2|φ2, φ2〉. (42)

This coherent state in H3 is then identified as the spin-1 state in some direction ofR3, which becomes more apparent if we choose the parameterisation c1 = cos 1

2θ andc2 = sin 1

2θ eiϕ. Clearly |ψ,ψ〉 is normalised in the sense of (19) since |c1|2 + |c2|

2 = 1. Ifwe call θ = 0 the positive z-direction in R3, then the triplet of states(

|φ1, φ1〉,|φ1, φ2〉 + |φ2, φ1〉

√2

, |φ2, φ2〉

)corresponds to the three spin-1 eigenstates of Sz:(

|Sz = +1〉, |Sz = 0〉, |Sz = −1〉).

An arbitrary state of the spin-1 particle is therefore expressed as a liner combinationof these basis states.

This line of construction extends to all higher spin particles. Thus, for example,for a spin-3

2 system we form the coherent state

|ψ,ψ,ψ〉 = c31|φ1, φ1, φ1〉 +

√

3c21c2

(|φ1, φ1, φ2〉 + |φ1, φ2, φ1〉 + |φ2, φ1, φ1〉

√3

)+√

3c1c22

(|φ1, φ2, φ2〉 + |φ2, φ1, φ2〉 + |φ2, φ2, φ1〉

√3

)+ c3

2|φ2, φ2, φ2〉 (43)

in H4 associated with |ψ〉 ∈ H2, and identify the four states appearing here as thefour eigenstates of the spin operator, and so on.

The formulation presented here is somewhat unduly rigid in that if we definea 2 × 2 Hermitian matrix ηi j = 〈φi|φ j〉, then the Hermitian transition amplitudes—asopposed to the physical transition amplitudes specified by (21)—between the spineigenstates for all higher spins are entirely specified by the 2× 2 matrix {ηi j}. In otherwords, the biorthogonal system for all higher spin systems are fixed once we fix thatof the underlying spin- 1

2 system. This rigidity, however, can in fact be relaxed, onaccount of Proposition 1, which shows that Hilbert space vectors play less prominentrole than one might have thought. In particular, in biorthogonal quantum mechanicsa coherent state can be constructed from incoherent Hilbert space vectors that arenevertheless projectively coherent. Thus, if |ψ〉 = c1|φ1〉 + c2|φ2〉 is given as beforeand if we define |ψ′〉 = c1|φ′1〉 + c2|φ′2〉, where 〈φi|φ j〉 , 〈φ′i |φ

′

j〉 so that |ψ〉 and |ψ′〉 are


inequivalent Hilbert space vectors, then we can still form an admissible coherent stateaccording to |ψ,ψ′〉. This follows on account of the fact that 〈χk|ψ〉 = 〈χ′k|ψ

′〉, k = 1, 2,

hence |ψ〉 and |ψ′〉 are projectively equivalent under our scheme. In this way we seethat the biorthogonal basis for each spin particle can be chosen arbitrarily, withoutconstraints.

The observation made in the previous paragraph also shows that in biorthogonalquantum theory an arbitrary pair of systems can be combined without constraints.This, in turn, clarifies one of the outstanding issues of combined systems in PT-symmetric quantum mechanics, which we shall discuss later. For now it sufficesto note that if one system represented by a Hilbert space H and another systemrepresented by a Hilbert spaceH ′ are combined, then the state vector of the combinedsystem is an element of the tensor product spaceH⊗H ′, just as in Hermitian quantummechanics. Thus, for example, if |ψ〉 = c1|φ1〉+ c2|φ2〉 is the state of one spin-1

2 particle,and |ψ′〉 = c′1|φ

′

1〉 + c′2|φ′

2〉 is the state of another such particle, then a disentangledproduct state inH ⊗H ′ takes the form

|ψ,ψ′〉 = c1c′1|φ1, φ′

1〉 + c1c′2|φ1, φ′

2〉 + c2c′1|φ2, φ′

1〉 + c2c′2|φ2, φ′

2〉, (44)

whereas a typical entangled state, such as the spin-0 singlet state, will be given by

|S = 0,Sz = 0〉 =1√

2

(|φ1, φ

′

2〉 − |φ2, φ′

1〉). (45)

This might appear paradoxical at first, since the singlet state has to be antisymmetric,which is not immediately apparent from the right side of (45). Indeed, |φn〉 and |φ′n〉represent distinct states in H , however, they are projectively equivalent, which inturn makes (45) antisymmetric in the projective Hilbert space.

For a combined system, the interaction Hamiltonian can also be represented ina manner analogous to that in standard quantum mechanics. Thus, in the case of apair of biorthogonal systems represented by a pair of Hamiltonians K = σx − iγσz andK′ = σx − iγ′σz with γ2, γ′2 < 1, the quantum Ising spin-spin interaction Hamiltoniancan be expressed in the form

σγz ⊗ σγ′

z =1√

(1 − γ2)(1 − γ′2)

1 iγ′ iγ −γγ′

iγ′ −1 −γγ′ −iγiγ −γγ′ −1 −iγ′

−γγ′ −iγ −iγ′ 1

, (46)

whose eigenvalues are, of course, given by (1,−1, 1,−1), independent of γ, γ′.

7. Perturbation analysis

We shall now turn to the perturbation analysis involving complex Hamiltonians, inthe range where there are no degeneracies so that the Rayleigh-Schrodinger series isapplicable. There is a substantial literature on perturbation theory involving complexHamiltonians, even in the vicinities of degeneracies where not only eigenvalues but


also eigenstates can be degenerate (see, for example, [16, 17, 28, 29, 30]). As such, wehave little new to add in this section, except perhaps the discussion on the nature ofthe operator that generates the perturbation, which turns out not to be unitary.

Let K be a complex Hamiltonian with distinct eigenvalues {κn} and biorthonormaleigenstates ({|φn〉}, {|χn〉}) that are known. Suppose that we perturb the Hamiltonianslightly according to

K→ Kε = K + εK′, (47)

where ε � 1 is the perturbation parameter, and K′ represents perturbation energy,which may or may not be Hermitian. Under the assumption that there areno degeneracies, the eigenstates {|ψn〉} and the eigenvalues {µn} of the perturbedHamiltonian Kε can be expanded in a power series

|ψn〉 = |φn〉 + ε|ψ(1)n 〉 + ε

2|ψ(2)

n 〉 + · · · , µn = κn + εµ(1)n + ε2µ(2)

n + · · · . (48)

As for the normalisation of the perturbed eigenstates, we shall assume that

〈χn|ψn〉 = 1. (49)

Since 〈χn|φn〉 = 1, it follows that under this normalisation convention we require

〈χn|ψ(1)n 〉 = 〈χn|ψ

(2)n 〉 = · · · = 0. (50)

It also means that 〈ψn|ψn〉 , 1, but the deviation from unity is negligible for ε� 1.If we substitute the series expansion (48) in the eigenvalue equation

Kε|ψn〉 = µn|ψn〉 (51)

and equate terms of different orders in ε, then we obtain

(κn − K)|φn〉 = 0, (κn − K)|ψ(1)n 〉 + µ

(1)n |φn〉 = K′|φn〉, (52)

and so on. Transvecting 〈χm| from the left on the second equation of (52) we obtain

(κn − κm)〈χm|ψ(1)n 〉 + µ

(1)n δnm = 〈χm|K′|φn〉. (53)

Thus, for n = m we obtain the first-order perturbation correction to the eigenvalue:

µ(1)n = 〈χn|K′|φn〉. (54)

On the other hand, for n , m we obtain

〈χm|ψ(1)n 〉 =

1κn − κm

〈χm|K′|φn〉, (55)

and on account of the completeness condition we thus find

|ψ(1)n 〉 =

∑m

|φm〉〈χm|ψ(1)n 〉 =

∑m,n

|φm〉〈χm|ψ(1)n 〉 =

∑m,n

〈χm|K′|φn〉

κn − κm|φm〉, (56)

where we have made use of the orthogonality relations (50). The results of [17]reproduced here for the first-order perturbation expansion lends itself naturally withthe analysis of geometric phases for complex Hamiltonians [31, 32, 33, 34].

It should be evident that higher-order perturbation corrections can be obtainedin a manner analogous to the standard perturbation theory in Hermitian quantum


mechanics, except the obvious modifications involving the biorthogonal basiselements. An important difference between (56) and the conventional result, however,is that instead of the orthogonality condition 〈φn|ψ

(1)n 〉 = 0, here we have 〈χn|ψ

(1)n 〉 = 0.

Now suppose that we regard Kε for |ε| � 1 as a one-parameter family of Hamiltoniansconnected to, and in the vicinity of, K. Then the eigenstates |ψn〉 for a small range ofε constitutes a segment of a path inH . If K is Hermitian, then a small displacementalong the path is unitary, and leaves the norm of the eigenstate invariant. In thepresent context, the displacement is generated by the operator

V =∑

n

|ψn(ε)〉〈χn|, (57)

where we have written |ψn(ε)〉 to make the ε dependence more explicit. In otherwords, we have V|φn〉 = |ψn〉. Evidently, V is not unitary, and hence its generatori(∂εV)V−1 is not Hermitian. In particular, perturbation of an eigenstate |φn〉 of acomplex Hamiltonian K does not leave the Dirac norm 〈φn|φn〉 of the state invariant,but instead leaves invariant the biorthogonal norm 〈χn|φn〉 of the state, and this inturn gives another support for the use of (21) as determining the physical probabilityrules involving complex Hamiltonians.

We remark, incidentally, that in the case of a Hermitian operator, a theorem ofRellich implies that the eigenstates and eigenvalues can be expanded in a Taylorseries of the form (48). However, for a general complex operator, the foregoingperturbation expansion breaks down in the vicinities of degeneracies where not onlythe eigenvalues but also the corresponding eigenstates coalesce. Such degeneraciesare often referred to as ‘exceptional points’ in the literature (see [35] and referencescited therein), with nontrivial observational consequences [36, 37]. Although theformal series expansion (48) breaks down in the neighbourhood of an exceptionalpoint, a perturbative analysis can nevertheless be pursued by employing the Newton-Puiseux series ([29], Theorem XII.2, [38]), as employed, e.g., in [21, 39, 40].

8. Dynamics

Thus far we have been considering static aspects of the eigenvalues and eigenstates ofa complex Hamiltonian K. We shall now turn to the analysis of the time evolution of aquantum state generated by such K, in the context of time-independent Hamiltonians.Specifically, we consider properties of the evolution operator

U = e−iKt, (58)

in units ~ = 1. Evidently, U is not unitary: U†U , 1. However, as we shall show, if theeigenvalues of K are real, then U in effect is unitary in the sense of biorthogonalquantum mechanics so that the norms of states and transition probabilities arepreserved under the time evolution.

It should be apparent that the solution to the dynamical equation

i∂t|ψ〉 = K|ψ〉, (59)


with initial condition |ψ0〉 =∑

n cn|φn〉, is given by

|ψt〉 =∑

n

cne−iκnt|φn〉. (60)

According to our conjugation rule (17) we thus have

〈ψt| =∑

n

cneiκnt〈χn| ⇒ |ψt〉 =

∑n

cne−iκnt|χn〉. (61)

The time-dependent biorthogonal norm of the state therefore is given by

〈ψt|ψt〉 =∑

n

cncne−i(κn−κn)t. (62)

We thus see that if the eigenvalues of K are real so that κn = κn, then for all time t > 0we have 〈ψt|ψt〉 = 〈ψ0|ψ0〉. More generally, if κn = κn, and if |ϕt〉 is also a solution tothe Schrodinger equation (59) with a different initial condition, then we have

〈ϕt|ψt〉 = 〈ϕ0|ψ0〉 (63)

for all t > 0. It follows that:

Proposition 2 If the eigenvalues of K are real, then the time evolution operator e−iKt isunitary with respect to the biorthogonal basis of K, preserving the biorthogonal norms of thestates and the transition probabilities between states.

Additionally, if the eigenvalues {κn} are real, then |ψt〉 can be seen to satisfy theSchrodinger equation i∂t|ψ〉 = K†|ψ〉 with the Hermitian-conjugated Hamiltonian K†.This, however, is not generally true if at least one of the eigenvalues of K is not real:i∂t|ψ〉 , K†|ψ〉 in general, which can be seen from (61).

When one or more of the eigenvalues are imaginary or complex, then we havedifferent characteristics for the dynamical behaviour of a quantum state. Let us write

κn = En − iγn (64)

for the eigenvalues, where {En} and {γn} are real. Then we have

〈ψt|ψt〉 =∑

n

cncne−2γnt = cn∗cn∗e−2γn∗ t

1 +∑n,n∗

cncn

cn∗cn∗e−2(γn−γn∗ )t

, (65)

where n∗ is the value of n such that γn has the smallest value (amongst the terms in theexpansion for which cn , 0). In most physical setups, γn ≥ 0, and an arbitrary initialstate will decay into the state with the smallest γn value, while at the same time theoverall norm decays. This situation describes the behaviour of a particle trapped in afinite potential well; the norm 〈ψt|ψt〉 then describes the probability that the particlehas not tunnelled out of the well. Note that if we let cn = δnk in (65) for some k, thenwe see that an eigenstate |φk〉 of K for which γk , 0 is not a stationary state, i.e. if|ψ0〉 = |φk〉, then 〈ψt|ψt〉 = e−2γkt.

The fact that when the eigenvalues are complex the state with the slowest decaywill in time dominate is of course well known in the context of systems with decays,


but it is worth remarking that as a consequence when such a system is immersedin a heat bath, it cannot result in an equilibrium configuration characterised by thethermal state (40).

With the notion of dynamics we are in a position to discuss time reversibility.In standard quantum mechanics there is no “one-size fits all” notion of the action oftime reversal operator (cf. [41]). Furthermore, the action of time reversal operatoris sometimes viewed as an antilinear map (a quadratic form) from the Hilbert spaceto its dual space: H → H ∗; and sometimes as an antilinear map (an operator) fromHilbert space to itself: H → H . Here we shall consider the latter convention, in linewith [42]. With the aid of a time-reversal operator T we can establish, for example,the following geometric identity

〈φm|φn〉 = 〈χn|χm〉 (66)

using the physical argument analogous to that presented in [17]. Suppose that we leta state evolve in time under the Hamiltonian K. From (65) the decay rate of |φn〉 isgiven by 2γn, whereas from (3) we have

γn =〈φn|Γ|φn〉

〈φn|φn〉. (67)

In other words, the decay rate of |φn〉 is determined by Γ (even though γn is not thephysical expectation of Γ in the state |φn〉). Since the time-reversed dynamics mustbe such that the state |φn〉 grows at the same rate 2γn, it follows that the time reversaloperatorT reverses the sign of iΓ but leaves H and Γ invariant: T KT −1 = K†. In otherwords, K†T = T K. Hence if we define

|χn〉 = T |φn〉, (68)

we find that |χn〉 is the eigenstate of K† with eigenvalue κn. The identity (66) thenfollows at once.

9. Relation to PT symmetry

As we have indicated earlier, interests in the study of classical and quantum systemsdescribed by complex, non-Hermitian Hamiltonians have increased significantlysince the realisation by Bender and Boettcher [43] that a wide class of complexHamiltonians possessing certain anti-linear symmetries can have entirely realeigenvalues. Specifically, the anti-linear symmetry considered in this context is thatassociated with the space-time inversion, i.e. parity-time (PT) reversal operation.Since the literature in the area of PT-symmetric quantum theory is substantial, andsince some of the ideas relating to biorthogonal quantum mechanics outlined herehave been identified directly or indirectly in the investigation of PT symmetry [9], itwill be useful to draw a special attention to the subject here.


We begin this discussion by recalling that, if we write 1 = (uu†)−1, then on accountof (30) we have

〈en|en〉 = 〈φn|1|φn〉 = 1 (69)

for all n, where 1 by construction is an invertible positive Hermitian operator, whichis unique and can be determined from the eigenstates [4]:

1−1 =∑

n

|φn〉〈φn|. (70)

In addition, observe, for all n, that

〈φn|12|φn〉 = 〈en|(u−1)†u−1(u−1)†u−1

|en〉 = 〈en|u−1(u−1)†|en〉 = 〈χn|χn〉, (71)

but (66) shows that 〈χn|χn〉 = 〈φn|φn〉, so that 1 is an involution:

12 = 1. (72)

Perceived from the viewpoint of Hermitian inner-product space, therefore, theoperator 1 plays the role of a ‘metric’ for the Hilbert space. For example, theexpectation value of a physical observable F can be written in the form

〈ψ|F|ψ〉〈ψ|ψ〉

=〈ψ|1F|ψ〉〈ψ|1|ψ〉

(73)

that involves the metric operator under the Hermitian pairing.We see therefore that biorthogonal quantum mechanics can alternatively be

viewed as ‘conventional’ Hermitian quantum mechanics, but where Hilbert spaceis endowed with a nontrivial metric operator 1. As remarked in §1, there are indeedproposals to equip Hilbert space with a nontrivial metric [1, 2]. The statement ofProposition 1, however, shows that for a physical system modelled on a finite-dimensional Hilbert space with a family of observables having real eigenvalues, thereare no observable consequences associated with the choice of the metric 1. Since anychoice of 1 is admissible, the Euclidean metric 1 = 1 seems to be the most economicalchoice, leading to standard quantum mechanics with Hermitian observables. Thus,possible physical significances of the metric 1, or equivalently biorthogonal quantummechanics, in a unitary system, can only be sought in infinite-dimensional systems.

The introduction of a nontrivial metric operator in Hilbert space emergedindependently in the context of PT-symmetric quantum mechanics [44, 45]. If aHamiltonian K is symmetric under the simultaneous parity-time inversion, then thefact that K possesses an anti-linear symmetry implies that its eigenvalues can be real.The parity operator P, however, cannot be used as a metric since it is not positive.Nevertheless, associated with such a Hamiltonian is another symmetry C, whoseproperties resemble those of a charge operator in quantum field theories, such that1 = CP can be used as a metric for Hilbert space [44, 45].

As a simple example, consider the class of Hamiltonians that are both symmetricand PT symmetric. The time-reversal operation considered in the literature of PTsymmetry is usually identified as the operation of complex conjugation. As regard


parity reversal, in the case of a system modelled on a finite-dimensional Hilbert spacethere is a priori no such notion of space reflection, and there is a freedom in the choiceof the parity operator. A canonical choice, however, is a finite-dimensional analogueof the space inversion operator, which is a counter-diagonal matrix whose counter-diagonal elements are all unity. With respect to a choice of orthonormal basis {|en〉}

we can thus write the parity operator P in the form:

P =∑

n

|en〉〈eN+1−n|, (74)

where N is the dimension of the Hilbert space. If the Hamiltonian K is symmetric,then we have

K =∑n,m

∑k,l

Kkluknul

m

|en〉〈em|. (75)

Thus, if we define time reversal to mean complex conjugation, we have

KPT =∑n,m

∑k,l

KklukN+1−nul

N+1−m

|en〉〈em|. (76)

The condition of PT symmetry, however, does not guarantee the reality of theeigenvalues. Nevertheless, if, in addition, the eigenstates {|φn〉} of K are also PTsymmetric, then we have uk

n = ukN+1−n. It follows that if a symmetric Hamiltonian K is

also PT symmetric, and if the eigenstates of K are likewise PT symmetric, then {Knm} arenecessarily real and symmetric (although the matrix elements of K in an orthonormalbasis are not real) so that the eigenvalues of K are real. Finally, conjugation operationcan be defined with the aid of

C =∑

n

(−1)n|φn〉〈χn|, (77)

such that 1 = CP defines the Hilbert space metric operator.One question that arises naturally in this context concerns the combined systems.

If one system is characterised by the metric operator 1, and another by 1′, can onecombine these systems in a meaningful way, and if so, how? Viewed as a systemcharacterised by a metric space, the canonical answers to these questions are notimmediately apparent; however, viewed as a biorthogonal quantum system, theformulation outlined in §6 provides a canonical way of treating combined systemsin this context. In particular, the metric operator for the combined system can beconstructed from the biorthogonal basis elements of the tensor-product space.

Interests in systems characterised by PT symmetry have increased significantlyover the past decade due to the observation that PT symmetry can be realised inlaboratories by balancing gain and loss. Based on the formal equivalence of paraxialapproximation to the scalar Hermholtz equation and the Schrodinger equation (see,e.g., [46, 47]), first experimental realisations of PT-symmetric systems were achieved inoptical waveguides [48]. Many other experiments have subsequently been proposed


or realised [49, 50, 51, 52, 53, 54, 55], although it should be added that experimentsthat have been realised so far involve classical systems, where measured quantitiesdo not correspond to eigenvalues of an observable acting on states of Hilbert space.

Quantum mechanically, the implication of the statement of Proposition 1 onPT symmetry is that whether a system is in complete isolation in the sense that allphysical observables are Hermitian, or whether the system is linked to an environmentsuch that gain and loss are balanced to the extent that all eigenmodes are PTsymmetric, an observer cannot detect any difference in the behaviour of the system.An interesting feature of PT-symmetric systems, however, is that most of the modelHamiltonians considered in the literature admit a tuneable parameter (or a set oftuneable parameters) such that even though the Hamiltonian K is PT symmetric,there are regions in parameter space where the eigenstates of K are not PT symmetric.In other words, the system admits two distinct phases (cf. [56]) associated withbroken and unbroken PT symmetry, and at the transition point the eigenstates ofK become degenerate (hence constitutes an example of an exceptional point). Thatthe eigenstates are degenerate implies that they lose the privileged status of beingcomplete; it follows from (30) that the operator u is not invertible, and consequentlythe metric operator 1 ceases to exist. Hence an experimental detection of a PT phasetransition in a purely quantum system modelled on a finite-dimensional Hilbert spacewill imply that physics beyond Hermitian Hamiltonians is not merely an intellectualcuriosity but rather is a requirement for the description of observed phenomena evenin the unitary contexts.

10. Discussion: towards infinite dimensional systems

The foregoing material has been based entirely on finite-dimensional aspects ofbiorthogonal quantum mechanics. It should be noted that already in quantummechanics based on conventional Hermitian operators there are subtleties in goingfrom finite to infinite-dimensional Hilbert spaces, and it should be intuitively clearthat the matter does not improve when considering quantum mechanics beyondHermitian operators. Thus, it will be neither feasible nor realistic to attempt to developa comprehensive account of biorthogonal quantum theory of infinite-dimensionalsystems here. Indeed, the following simple example of Young [57] already illustrateshow a completeness statement of biorthogonal quantum mechanics that holds truein finite dimensions can easily fail in infinite dimensions.

Consider an infinite-dimensional Hilbert spaceH and an orthonormal set of basis{|en〉} inH . Construct a new set of basis elements {|φn〉} according to the prescription

|φn〉 = |e1〉 + |en〉 (78)

for n = 2, 3, . . . ,∞. Evidently, elements of {|φn〉} are not orthogonal, but the set is


nonetheless complete since

limN→∞

1N − 1

N∑n=2

|φn〉 = |e1〉 + limN→∞

1N − 1

N∑n=2

|en〉 = |e1〉, (79)

on account of the fact that the term orthogonal to |e1〉 in the left side of (79) decays atthe rate (N − 1)−1/2. It should be evident that the biorthogonal pair of |φn〉 is uniqueand is given by

|χn〉 = |en〉 (80)

for n = 2, 3, . . . ,∞, so that we have 〈χn|φm〉 = δnm. While the set {|φn〉} is complete, itsbiorthogonal counterpart {|χn〉} is not—a phenomenon that has no analogue in finitedimensions. Thus, if K =

∑n κn|φn〉〈χn| is a Hamiltonian operator acting on the states

ofH , then we can form a linear combination of the eigenstates of K according to (79)that has a null conjugate state:

〈e1|e1〉 = 0. (81)

If we interpret the norm as representing the probability of finding a particle in thesystem, then we have a ‘no-particle’ state |e1〉 that nevertheless has nonzero energyexpectation value, since 〈K〉 in the state |e1〉 is formally given by the uniform averageof the energy eigenvalues, which may be finite or infinite, but will be nonzero.

Even if a biorthonormal set ({|φn〉}, {|χn〉}) is complete, there can be varioussubtleties arising from the lack of a bounded map that takes an element |φn〉 into|en〉. Specifically, suppose that ({|φn〉}, {|χn〉}) is a complete biorthonormal set of basesin the Hilbert space H = L2 of square-integrable functions. Then the set {|φn〉} iscalled a ‘Fischer-Riesz’ basis if (a) for any |ψ〉 ∈ H we have

∑n |〈χn|ψ〉|2 < ∞; and

(b) if for any sequence {cn} such that∑

n |cn|2 < ∞ there exists a |ψ〉 ∈ H for which

〈χn|ψ〉 = cn. A theorem of Bari [14] then shows that: (i) {|χn〉} is a Fischer-Riesz basisif and only if there exists a bounded invertible linear operator u−1 and a completeorthonormal basis elements {|en〉} in H such that u−1

|φn〉 = |en〉; and that (ii) {|φn〉} isa Fischer-Riesz basis if and only if there exists a positive bounded invertible linearoperator 1−1 inH such that |φn〉 = 1−1

|χn〉.In §9 we have shown that these results are easily verified in the case of a finite-

dimensional Hilbert space. In infinite dimensions, on the other hand, a genericcomplex Hamiltonian K possessing real eigenvalues often do not admit an invertiblebounded metric operator 1, and this implies that a system described by such aHamiltonian is intrinsically different from that described by a Hermitian Hamiltonian,even if the eigenvalues coincide. There is an active research into identifying variousimplications of the lack of such metric operators in various systems [58, 59, 60, 61, 62],however, observable effects relating to these subtleties have yet to be identified.

In conclusion, let us summarise the main message of the paper. In the case ofquantum systems modelled on finite-dimensional Hilbert spaces, provided that anoperator possesses real eigenvalues and a complete set of eigenstates, it is a viablecandidate to represent a physical observable, irrespective of whether it is Hermitian


in the conventional sense. In particular, there seems to be no experiment that one canperform to determine overlap distances between the eigenstates in a Hilbert spaceH ,since nonorthogonal eigenstates inH nevertheless correspond to orthogonal states inthe projective Hilbert space, in the framework of biorthogonal (and unitary) quantummechanics. The situation, of course, changes if one is characterising manifestly openquantum systems lacking unitarity, for which one or more of the eigenvalues arenot real (see, e.g., [63] for a discussion on the determination of the Petermann factor[〈χn|φn〉〈φn|χn〉/〈χn|χn〉〈φn|φn〉]−1 in an optical cavity, or [30] for a discussion on thedetection of the lack of orthogonality from the statistics of resonance widths).

Whether the same conclusion concerning the lack of identifiability of theorthogonality of states in a unitary theory extends into infinite-dimensional Hilbertspaces remains an open question. In this case, the wave function encodes informationconcerning the configuration of the space in which particles exist, in the form ofasymptotic boundary conditions. For example, for a one-dimensional system, thewave function may be defined on the real line, or along a contour in the complex plane(such as the PT-symmetric negative quartic potential [43]), depending on the relevantboundary conditions. Since any such contour can lie along the real axis in a regionthat is experimentally relevant, it is not a priori clear whether local measurementsperformed in this region can determine if the wave function should decay along astraight line or along a curve at infinities.

Acknowledgements

The author acknowledges Eva-Maria Graefe, Bernhard Meister, and Matthew Parryfor stimulating discussion, and the two anonymous referees for helpful comments.

References

[1] Scholtz, F. G., Geyer, H. B. & Hahne, F. J. W. Quasi-Hermitian operators in quantum mechanicsand the variational principle. Annals of Physics 213, 74-101 (1992).

[2] Geyer, H. B., Heiss, W. D. & Scholtz, F. G. The physical interpretation of non-HermitianHamiltonians and other observables. Canadian Journal of Physics 86, 1195-1201 (2008).

[3] Bender, C. M. Making sense of non-Hermitian Hamiltonians. Reports on Progress in Physics 70,947-1018 (2007).

[4] Mostafazadeh, A. Pseudo-Hermitian representation of quantum mechanics. International Journal ofGeometric Methods in Modern Physics 7, 1191-1306 (2010).

[5] Znojil, M. Three-Hilbert-space formulation of quantum mechanics. Symmetry, Integrability andGeometry: Methods and Applications 5, 001 (2009).

[6] Moiseyev, N. Quantum theory of resonances: calculating energies, widths and cross-sections bycomplex scaling. Physics Reports 302, 211-293 (1998).

[7] Okołowicz, J., Płoszajczak, M. & Rotter, I. Dynamics of quantum systems embedded in a continuum.Physics Report 374 271-383 (2003).

[8] Moiseyev, N. Non-Hermitian Quantum Mechanics. (Cambridge: Cambridge University Press, 2011).[9] Curtright, T. & Mezincescu, L. Biorthogonal quantum systems. Journal of Mathematical Physics 48,

092106 (2007).


[10] Liouville, J. Premier Memoire sur la Theorie des Equations differentielles linaires et sur ledeveloppement des Fonctions en series. Journal de Mathematiques Pures et Appliquees 2, 561-614(1838).

[11] Birkhoff, G. D. Boundary value and expansion problems of ordinary linear differential equations.Transactions of the American Mathematical Society 9, 373-395 (1908).

[12] Pell, A. J. Biorthogonal systems of functions. Transactions of the American Mathematical Society 12,135-164 (1911).

[13] Pell, A. J. Applications of biorthogonal systems of functions to the theory of integral equations.Transactions of the American Mathematical Society 12, 165-180 (1911).

[14] Bari, N. K. Biorthogonal systems and bases in Hilbert space. Moskov. Gos. Univ. Uenye ZapiskiMatematika 148, 69-107 (1951).

[15] Gelfand, I. M. Remark on the work of N. K. Bari “Biorthogonal systems and bases in Hilbertspace”. Moskov. Gos. Univ. Uenye Zapiski Matematika 148, 224-225 (1951).

[16] More, R. M. Theory of decaying states. Physical Review A4, 1782-1790 (1971).[17] Sternheim, M. M. & Walker, J. F. Non-Hermitian Hamiltonians, decaying states, and perturbation

theory. Physical Review C6, 114-121 (1972).[18] des Cloizeaux, J. Energy bands and projection operators in a crystal: Analytic and asymptotic

properties. Physical Review 135, A685-A697 (1964).[19] des Cloizeaux, J. Extension d’une formule de Lagrange a des problemes de valeurs propres.

Nuclear Physics 20, 321-346 (1960).[20] Keck, F., Korsch, H. J. & Mossmann, S. Unfolding a diabolic point: a generalised crossing scenario.

Journal of Physics A36, 2125-2137 (2003).[21] Gunther, U., Rotter, I. & Samsonov, B. F. Projective Hilbert space structures at exceptional points.

Journal of Physics A40, 8815-8833 (2007).[22] Brody, D. C. & Graefe, E. M. Six-dimensional space-time from quaternionic quantum mechanics.

Physical Review D84, 125016 (2011).[23] Muller, M. P. & Masanes, L. Three-dimensionality of space and the quantum bit: an information-

theoretic approach. New Journal of Physics 15, 053040 (2013).[24] Dakic, B. & Brukner, C. The classical limit of a physical theory and the dimensionality of space.

In Quantum Theory: Informational Foundations and Foils, G. Chiribella and R. W. Spekkens, eds.(Berlin: Springer, 2013).

[25] Friedrich, B. & Herschbach, D. Stern and Gerlach: How a bad cigar helped reorient atomic physics.Physics Today 56, 53-59 (2003).

[26] Brody, D. C. & Hughston, L. P. Geometric quantum mechanics. Journal of Geometry and Physics 38,19-53 (2001).

[27] Brody, D. C. & Graefe, E. M. Coherent states and rational surfaces. Journal of Physics A43, 255205(2010).

[28] Kato, T. Perturbation Theory for Linear Operators, 2nd ed. (Berlin: Springer-Verlag, 1976).[29] Reed, M. & Simon, B. Methods of Modern Mathematical Physics IV: Analysis of Operators (New York:

Academic Press, 1978).[30] Fyodorov, Y. V. & Savin, D. V. Statistics of resonance width shifts as a signature of eigenfunction

nonorthogonality. Physical Review Letters 108, 184101 (2012).[31] Garrison, J. C. & Wright, E. M. Complex geometrical phases for dissipative systems. Physics Letters

A128, 177-181 (1988).[32] Mailybaev, A. A., Kirillov, O. N. & Seyranian, A. P. Geometric phase around exceptional points.

Physical Review A72, 014104 (2005).[33] Mehri-Dehnavi, H. & Mostafazadeh, A. Geometric phase for non-Hermitian Hamiltonians and

its holonomy interpretation. Journal of Mathematical Physics 49, 082105 (2008).[34] Cui, X. D. & Zheng, Y. Geometric phases in non-Hermitian quantum mechanics. Physical Review

A72, 064104 (2012).[35] Heiss, W. D. The physics of exceptional points. Journal of Physics A45, 444016 (2012).


[36] Dembowski, C., Graf, H. D., Harney, H. L., Heine, H. L., Heiss, W. D., Rehfeld, H. & Richter, A.Experimental observation of the topological structure of exceptional points. Physical ReviewLetters 86, 787-790 (2001).

[37] Dembowski, C., Dietz, B., Graf, H. D., Harney, H. L., Heine, H. L., Heiss, W. D. & Richter, A.Observation of a chiral state in a microwave cavity. Physical Review Letters 90, 034101 (2003).

[38] Seyranian, A. P. & Mailybaev, A. A. Multiparameter Stability Theory with Mechanical Applications.(Signapore: World Scientific, 2003).

[39] Graefe, E. M., Gunther, U., Korsch, H. J. & Niederle, A. E. A non-Hermitian symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points. Journal ofPhysics A41, 255206 (2008).

[40] Demange, G. & Graefe, E. M. Signatures of three coalescing eigenfunctions. Journal of Physics A45,025303 (2012).

[41] Morpurgo, G., Touschek, B. F. & Radicati, L. A. On time reversal. Nuovo Cimento 12, 677-698 (1954).[42] Wigner, E. Uber die Operation der Zeitumkehr in der Quantenmechanik. Nachrichten von der

Gesellschaft der Wissenschaften zu Gottingen, Mathematisch-Physikalische Klasse 1932, 546-559 (1932).[43] Bender, C. M. & Boettcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry.

Physical Review Letters 80, 5243-5246 (1998).[44] Bender, C. M., Brody, D. C. & Jones, H. F. Complex extension of quantum mechanics. Physical

Review Letters 89, 270401 (2002).[45] Mostafazadeh, A. Pseudo-Hermiticity versus PT-symmetry III. Journal of Mathematical Physics 43,

3944-3951 (2002).[46] Steuernagel, O. Equivalence between focused paraxial beams and the quantum harmonic

oscillator. American Journal of Physics 73, 625-629 (2005).[47] Klaiman, S., Gunther, U. & Moiseyev, N. Visualization of branch points in PT-symmetric

waveguides. Physical Review Letters 101, 080402 (2008).[48] Guo, A., Salamo, G. J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V.,

Siviloglou, G. A. & Christodoulides, D. N. Observation of PT-symmetry breaking in complexoptical potentials. Physical Review Letters 103, 093902 (2009).

[49] Mostafazadeh, A. Spectral singularities of complex scattering potentials and infinite reflectionand transmission coefficients at real energies. Physical Review Letters 102, 220402 (2009).

[50] Schomerus, H. Quantum noise and self-sustained radiation of PT-Symmetric systems. PhysicalReview Letters 104, 233601 (2010).

[51] Ruter, C. E., Makris, K. G., El-Ganainy, R., Christodoulides, D. N. & Kip, D. Observation ofparitytime symmetry in optics. Nature Physics 6, 192-195 (2010).

[52] Liertzer, M., Ge, L., Cerjan, A., Stone, A. D., Tureci, H. E. & Rotter, S. Pump-induced exceptionalpoints in lasers. Physical Review Letters 108, 173901 (2012).

[53] Schindler, J., Li, A., Zheng, M. C., Ellis, F. M. & Kottos, T. Experimental study of active LRC circuitswith PT symmetries. Physical Review A84, 040101 (2011).

[54] Bittner, S., Dietz, B., Gunther, U., Harney, H. L., Miski-Oglu, M., Richter, A. & Schafer, F. PTsymmetry and spontaneous symmetry breaking in a microwave billiard. Physical Review Letters108, 024101 (2012).

[55] Bender, C. M., Berntson, B. K., Parker, D. & Samuel, E. Observation of PT phase transition in asimple mechanical system American Journal of Physics 81, 173-179 (2013).

[56] Brody, D. C. & Graefe, E. M. Mixed-state evolution in the presence of gain and loss. Physical ReviewLetters 109, 230405 (2012).

[57] Young, R. M. On complete biorthogonal systems. Proceedings of the Americal Mathematical Society83, 537-540 (1981).

[58] Kretschmer, R. & Szymanowski, L. Quasi-Hermiticity in infinite-dimensional Hilbert spaces.Physics Letters A325, 112-117 (2004).

[59] Bagarello, F. Mathematical aspects of intertwining operators: the role of Riesz bases. Journal ofPhysics A43, 175203 (2010).


[60] Siegl, P. & Krejcirık, D. On the metric operator for the imaginary cubic oscillator. Physical ReviewD86, 121702(R) (2012).

[61] Bender, C. M. & Kuzhel, S. Unbounded C symmetries and their nonuniqueness. Journal of PhysicsA43, 444005 (2012).

[62] Mostafazadeh, A. Pseudo-Hermitian quantum mechanics with unbounded metric operators.Philosophical Transactions of the Royal Society London A371, 20120050 (2013).

[63] Berry, M. V. Mode degeneracies and the Petermann excess-noise factor for unstable lasers. Journalof Modern Optics 50, 63-81 (2003).

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